Reducing Congestion Effects in Wireless Networks by Multipath Routing
Is Multipath Routing Really a Panacea?
Transcript of Is Multipath Routing Really a Panacea?
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Is Multipath Routing Really a Panacea?
Deep Medhi
Computer Science & Electrical Engineering DepartmentUniversity of Missouri-Kansas City, USA
[email protected] association with Xuan Liu, Sudhir Mohanraj, and Michał Pioro
Supported in part by NSF Grant # CNS-0916505
CNSM Keynote: 11 November 2015
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Keynote dedicatedin memory ofKaren Medhi
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Manhattan, NY
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Going between Two points in Manhattan, NY
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Going between Two points in Manhattan, NY: one path
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Going between Two points in Manhattan: two paths
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Going between Two points in Manhattan: three paths
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Going between Several points in Manhattan, NY
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Going between *ANY* two points in Manhattan, NY
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Question:
At instant of time, is multipath routing beneficial?
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What Multipath Routing is NOT
Take one path in the morning, another path in the evening – thisis NOT multipath routing
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Multipath Routing: A Common Belief
Multipath routing (load sharing): split routing where eachnode-to-node traffic can be send among accessible paths.
An alternative considered in this context is single-pathrouting (non-split routing of the demands).— Many ISPs prefer (for troubleshooting)Common belief: multipath routing, as compared withsingle-path routing, gives a significantly better opportunityto control the link loads and in this way effectively optimizevarious traffic objectives.Question: Does it? When? How much?Taking a Traffic Engineering Perspective...
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Multipath Routing: A Common Belief
Multipath routing (load sharing): split routing where eachnode-to-node traffic can be send among accessible paths.
An alternative considered in this context is single-pathrouting (non-split routing of the demands).— Many ISPs prefer (for troubleshooting)
Common belief: multipath routing, as compared withsingle-path routing, gives a significantly better opportunityto control the link loads and in this way effectively optimizevarious traffic objectives.Question: Does it? When? How much?Taking a Traffic Engineering Perspective...
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Multipath Routing: A Common Belief
Multipath routing (load sharing): split routing where eachnode-to-node traffic can be send among accessible paths.
An alternative considered in this context is single-pathrouting (non-split routing of the demands).— Many ISPs prefer (for troubleshooting)Common belief: multipath routing, as compared withsingle-path routing, gives a significantly better opportunityto control the link loads and in this way effectively optimizevarious traffic objectives.
Question: Does it? When? How much?Taking a Traffic Engineering Perspective...
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Multipath Routing: A Common Belief
Multipath routing (load sharing): split routing where eachnode-to-node traffic can be send among accessible paths.
An alternative considered in this context is single-pathrouting (non-split routing of the demands).— Many ISPs prefer (for troubleshooting)Common belief: multipath routing, as compared withsingle-path routing, gives a significantly better opportunityto control the link loads and in this way effectively optimizevarious traffic objectives.Question: Does it? When? How much?
Taking a Traffic Engineering Perspective...
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Multipath Routing: A Common Belief
Multipath routing (load sharing): split routing where eachnode-to-node traffic can be send among accessible paths.
An alternative considered in this context is single-pathrouting (non-split routing of the demands).— Many ISPs prefer (for troubleshooting)Common belief: multipath routing, as compared withsingle-path routing, gives a significantly better opportunityto control the link loads and in this way effectively optimizevarious traffic objectives.Question: Does it? When? How much?Taking a Traffic Engineering Perspective...
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A 3-node Example: Single Demand (Commodity)
1
3
2
10
10
10Capacity
15
Traffic Volumebetween 1 and 2
Easy to see that 15 units of traffic would need to be splitbetween the paths 1-2 and 1-3-2.
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3-node example: All pairs with traffic
1
3
2
Capaci
ty = 10Capacity = 10
Capacity = 15
Traffic = 5
Traffic = 7Traffic =
10
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Min Cost Routing: (i) Illustration for 3-node network: all pair traffic
Linear Programming Formulation:Minimize x12 + 2 x132 + x13 + 2 x123 + x23 + 2 x213
subject to
d12: x12 + x132 = 5d13: x13 + x123 = 10d23: x23 + x213 = 7c12: x12 + x123 + x213 <= 10c13: x132 + x13 + x213 <= 10c23: x132 + x123 + x23 <= 15Boundsx12 >= 0x132 >= 0x13 >= 0x123 >= 0x23 >= 0x213 >= 0End
1
3
2
Capaci
ty = 10
Capacity = 10
Capacity = 15
Traffic = 5
Traffic = 7Traffic =
10For each pair, the first (direct) path is cheaper than the second path.Solution: x12 = 5, x13 = 10, x23 = 7 3 positive path-flows
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3-node example: All pairs with traffic
1
3
2
Capaci
ty = 10
Capacity = 10Capacity = 15
Traffic = 5
Traffic = 7Traffic =
10
Is it possible for each pair to use both the direct and alternatetwo-link paths with positive flows at optimality?
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Min Cost Routing: (ii) 3-node net (swap path cost)
Minimize 2 x12 + x132 + 2 x13 + x123 + 2 x23 + x213subject to
d12: x12 + x132 = 5d13: x13 + x123 = 10d23: x23 + x213 = 7c12: x12 + x123 + x213 <= 10c13: x132 + x13 + x213 <= 10c23: x132 + x123 + x23 <= 15Boundsx12 >= 0x132 >= 0x13 >= 0x123 >= 0x23 >= 0x213 >= 0End
1
3
2
Capaci
ty = 10
Capacity = 10Capacity = 15
Traffic = 5
Traffic = 7Traffic =
10
Note: same traffic demand and link capacity as before; the path cost in objective ischanged. Solution: x12 = 1, x132 = 4, x13 = 3.5, x123 = 6.5, x23 = 4.5, x213 = 2.56 positive path-flows
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Is there a connection?
1) How does the number of positive flows at optimality relate tothe size of the problem?2) Does the objective function matter?
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Two Letters to Remember for the Rest of the Talk
D : Number of demands in a Network
L : Number of Links in a Network
NOTE:D = N(N − 1)/2 if every pair has traffic (bidirectional) in aN-node network
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Two Letters to Remember for the Rest of the Talk
D : Number of demands in a Network
L : Number of Links in a Network
NOTE:D = N(N − 1)/2 if every pair has traffic (bidirectional) in aN-node network
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D + L result: Min Cost Routing Multi-CommodityNetwork Flow
minx≥0
∑d∈D
∑p∈Pd
ξdp xdp (1a)
∑p∈Pd
xdp = hd , d ∈ D (demand) (1b)
∑d∈D
∑p∈Pd
δdp`xdp ≤ c`, ` ∈ L (capacity) (1c)
hd : traffic for demand ID d ∈ D (#(D) = D)c`: link capacity (#(L) = L)ξdp: unit path cost of path p for demand dδdp`: link-path indicator 0/1
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Min Cost Routing Multi-Commodity Network Flow
minx≥0
∑d∈D
∑p∈Pd
ξdp xdp∑p∈Pd
xdp = hd , d ∈ D (demand)
∑d∈D
∑p∈Pd
δdp`xdp ≤ c`, ` ∈ L (capacity)
D + L property
In vertex solutions in Linear Program, there are at most D + L positivepath-flows. (Proof skipped)
Corollary: There are at most L demands that require more than onepositive path-flow
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Quick Review: Feasible Region and vertices for a linear program
Objective
Feasible Region
Optimal Vertex
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Consider again the following illustration for the 3-nodemulti-commodity example:
d12: x12 + x132 = 5d13: x13 + x123 = 10d23: x23 + x213 = 7c12: x12 + x123 + x213 <= 10c13: x132 + x13 + x213 <= 10c23: x132 + x123 + x23 <= 15x non-negative
Adding slack variables:
d12: x12 + x132 = 5d13: x13 + x123 = 10d23: x23 + x213 = 7c12: x12 + x123 + x213 + s12 = 10c13: x132 + x13 + x213 + s13 = 10c23: x132 + x123 + x23 + s23 = 15x, s non-negative
Here, D = 3, L = 3. After adding slack variables, we have six (6) equations with nine(9) variables. This means that at most 6 variables can be positive at a basic feasiblesolution, and consequently, at optimality.
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CorollaryIf the optimization problem (1) is feasible, then at most L trafficpairs will have more than one path with non-zero flows atoptimality.
Proof:
From the theorem, we know that there are D + L non-zero flowvariables. Since there are a total of D pairs, at least one pathfor each pair must carry the traffic load. This then leaves uswith at most D + L− D = L pairs that has more than one pathswith non-zero flows.
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CorollaryIf the optimization problem (1) is feasible, then at most L trafficpairs will have more than one path with non-zero flows atoptimality.
Proof:
From the theorem, we know that there are D + L non-zero flowvariables. Since there are a total of D pairs, at least one pathfor each pair must carry the traffic load. This then leaves uswith at most D + L− D = L pairs that has more than one pathswith non-zero flows.
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Traffic Engineering Objectives (besides min costrouting)
Commonly used traffic engineering objectives for IP, MPLS,SDN networks:
Network Load Balancing (Minimize Maximum utilization),also known as Congestion MinimizationAverage Delay (Minimize Average Network Delay)
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Load Balancing Optimization: LP Formulation
min{x≥0,r}
r
subject to ∑p∈Pd
xdp = hd , d ∈ D∑d∈D
∑p∈Pd
δdp` xdp ≤ c`r , ` ∈ L
xdp ≥ 0, p = 1, 2, ...,Pd ,d = 1, 2, ...,D
(3)
Note: introduced a new variable r (load balancing variable)
We again have D + L constraints.
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LB: load balancing
minx≥0,r
r∑p∈Pd
xdp = hd , d ∈ D
∑d∈D
∑p∈Pd
δdp`xdp ≤ c`r , ` ∈ L
xdp ≥ 0, p = 1,2, ...,Pd , d = 1,2, ...,D
D + L − 1 property
In vertex solutions, there are at most D + L − 1 positive path-flows.
There are at most L − 1 demands that require more than one positivepath-flow.
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3-node Illustration for load balancing
Minimize rsubject to
d12: x12 + x132 = 5d13: x13 + x123 = 10d23: x23 + x213 = 7c12: x12 + x123 + x213 <= 10 rc13: x132 + x13 + x213 <= 10 rc23: x132 + x123 + x23 <= 15 rBoundsx12 >= 0x132 >= 0x13 >= 0x123 >= 0x23 >= 0x213 >= 0End
1
3
2
Capaci
ty = 10
Capacity = 10Capacity = 15
Traffic = 5
Traffic = 7Traffic =
10
Solution: x12 = 4.125, x132 = 0.875, x13 = 6.625, x123 = 3.375, x23 = 7, r∗ = 0.75D + L− 1 = 3 + 3− 1 = 5
One pair always has single-path routing at optimality
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3-node Illustration for load balancing
Minimize rsubject to
d12: x12 + x132 = 5d13: x13 + x123 = 10d23: x23 + x213 = 7c12: x12 + x123 + x213 <= 10 rc13: x132 + x13 + x213 <= 10 rc23: x132 + x123 + x23 <= 15 rBoundsx12 >= 0x132 >= 0x13 >= 0x123 >= 0x23 >= 0x213 >= 0End
1
3
2
Capaci
ty = 10
Capacity = 10Capacity = 15
Traffic = 5
Traffic = 7Traffic =
10
Solution: x12 = 4.125, x132 = 0.875, x13 = 6.625, x123 = 3.375, x23 = 7, r∗ = 0.75D + L− 1 = 3 + 3− 1 = 5 One pair always has single-path routing at optimality
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Minimize Average Delay (AD)
minx≥0,y≥0
∑∈L
y`c`−y`
subject to∑
p∈Pd
xdp = hd , d ∈ D∑d∈D
∑p∈Pd
δdp` xdp = y`, ` ∈ L(5)
Note: Objective is non-linear.
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D + L for non-linear AD
The D + L property holds for the non-linear AD problem!
minx≥0,y≥0
∑∈L
y`c`−y`
subject to∑
p∈Pd
xdp = hd , d ∈ D∑d∈D
∑p∈Pd
δdp` xdp = y`, ` ∈ L(6)
We’ve a proof!
Margins of this slide is too small to fit in the proof :)
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D + L for non-linear AD
The D + L property holds for the non-linear AD problem!
minx≥0,y≥0
∑∈L
y`c`−y`
subject to∑
p∈Pd
xdp = hd , d ∈ D∑d∈D
∑p∈Pd
δdp` xdp = y`, ` ∈ L(6)
We’ve a proof!
Margins of this slide is too small to fit in the proof :)
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Two Measures
What percentage of demand pairs have more than onepath at optimality?– MPM (Multipath Measure)
How far off is the single-path routing compared to multipathrouting?– Normalized Cost Overhead (COH)
COH =OPTSinglePath −OPTMultiPath
OPTMultiPath
NOTE: MPM = 0 −→ Single-Path Routing is Optimal
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Two Measures
What percentage of demand pairs have more than onepath at optimality?– MPM (Multipath Measure)How far off is the single-path routing compared to multipathrouting?– Normalized Cost Overhead (COH)
COH =OPTSinglePath −OPTMultiPath
OPTMultiPath
NOTE: MPM = 0 −→ Single-Path Routing is Optimal
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Two Measures
What percentage of demand pairs have more than onepath at optimality?– MPM (Multipath Measure)How far off is the single-path routing compared to multipathrouting?– Normalized Cost Overhead (COH)
COH =OPTSinglePath −OPTMultiPath
OPTMultiPath
NOTE: MPM = 0 −→ Single-Path Routing is Optimal
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MPM: multipath measure
definitionMPM is equal to the maximum percentage of demands that canhave more than one path with nonzero flow at optimal vertexsolutions.
MCR: MPM = LD
LB: MPM = L−1D
AD: MPM = LD
(max value = 100%)
MPM computed to optimality: to be denoted by MPM∗
MPM∗ ≤ MPM
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MPM: multipath measure
definitionMPM is equal to the maximum percentage of demands that canhave more than one path with nonzero flow at optimal vertexsolutions.
MCR: MPM = LD
LB: MPM = L−1D
AD: MPM = LD
(max value = 100%)
MPM computed to optimality: to be denoted by MPM∗
MPM∗ ≤ MPM
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Average vertex degree for Real-world ISP Networks
Topology Zoo Collection: the highest average vertex degree 4.5
RocketFuel Collection (PoP-level topology): the highest average vertex degree5.24
Use ”3N-Net” as an upper limit for theoretical MPM– L = 3N means average vertex degree is 6
Important to note: with L = O(N), D = O(N2)
MPM (=L/D) −→ 0 as N →∞
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Average vertex degree for Real-world ISP Networks
Topology Zoo Collection: the highest average vertex degree 4.5
RocketFuel Collection (PoP-level topology): the highest average vertex degree5.24
Use ”3N-Net” as an upper limit for theoretical MPM– L = 3N means average vertex degree is 6
Important to note: with L = O(N), D = O(N2)
MPM (=L/D) −→ 0 as N →∞
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Average vertex degree for Real-world ISP Networks
Topology Zoo Collection: the highest average vertex degree 4.5
RocketFuel Collection (PoP-level topology): the highest average vertex degree5.24
Use ”3N-Net” as an upper limit for theoretical MPM– L = 3N means average vertex degree is 6
Important to note: with L = O(N), D = O(N2)
MPM (=L/D) −→ 0 as N →∞
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Average vertex degree for Real-world ISP Networks
Topology Zoo Collection: the highest average vertex degree 4.5
RocketFuel Collection (PoP-level topology): the highest average vertex degree5.24
Use ”3N-Net” as an upper limit for theoretical MPM– L = 3N means average vertex degree is 6
Important to note: with L = O(N), D = O(N2)
MPM (=L/D) −→ 0 as N →∞
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Theoretical MPM
N = 4 N = 9 N = 16 N = 25 N = 36 N = 49 N = 100 N = 1024
D 6 36 120 300 630 1,176 4,950 523,776R(MCR/AD) 66.67 25.00 13.33 8.33 5.71 4.17 2.02 0.20R(LB) 50.00 22.22 12.50 8.00 5.56 4.08 2.00 0.20G(MCR/AD) 66.67 33.33 20.00 13.33 9.52 7.14 3.64 0.38G(LB) 50.00 30.56 19.17 13.00 9.37 7.06 3.62 0.383N(MCR/AD) 100.00 75.00 40.00 25.00 17.14 12.50 6.06 0.593N(LB) 83.33 72.22 39.17 24.67 16.98 12.41 6.04 0.59
MPM (in %) for different network sizes and topologies
D = N/(N-1)/2 (No. of demand pairs)R(Ring) L = N
G(Grid) L = 2N − 2√
N3N(3N-Net) L = 3N
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Special Results: Ring Network, uniform traffic/capacitycase (”Symmetric Ring”): LB Objective
In a ring network, every pair of nodes have two paths:clockwise and counter-clockwise.
If the number of nodes N is odd, then in a ring with uniformtraffic for all node pairs and link capacity (”symmetric ring”),only one path (minimum-hop shortest path) is used byeach pair at optimality for the LB objective. That is,MPM∗ = 0If the number of nodes N is even, then in a symmetric ringMPM∗ = 1
N−1
Same hold for AD objective too.
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Special Results: Ring Network, uniform traffic/capacitycase (”Symmetric Ring”): LB Objective
In a ring network, every pair of nodes have two paths:clockwise and counter-clockwise.If the number of nodes N is odd, then in a ring with uniformtraffic for all node pairs and link capacity (”symmetric ring”),only one path (minimum-hop shortest path) is used byeach pair at optimality for the LB objective. That is,MPM∗ = 0If the number of nodes N is even, then in a symmetric ringMPM∗ = 1
N−1
Same hold for AD objective too.
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Symmetric Ring - Odd number of nodes - Illustration(N = 5)
12
3
4
5
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Symmetric Ring - Even number of nodes - Illustration(N = 4)
1 2
34
No split for: 1:2, 2:3, 3:4, 1:4 greenequal split for: 1:3 (blue), 2:4 (brown)MPM∗ = 2/6 = 1/3
If N = 101, then MPM∗ = 1/100 = 1%
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A Series of Comprehensive Studies
Topologies: Ring, Grid, Example ISP topologiesTraffic distribution: uniform (U), uniform-perturbed (U-P),Elephant-mice (EM) traffic, Lognormal (LN) trafficNetwork Load: 0.4 to 0.95For each load, five traffic profiles generated randomlyFor Load Balancing Objective, we proved a traffic scalingproperty that the optimal solution doesn’t change.A few representative results presented here.
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A Representative set of resultsCases
All pair traffic (D = N(N − 1)/2)Limited pair traffic for Data Center NetworksWhat happens as we increase D from 1,2, ....,N(N − 1)/2
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Ring Topology: MPM∗ and COH
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Grid Topology: MPM∗ and COH
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Fully-Mesh Topology
Theoretical MPM: 100%
Symmetric Mesh: Uniform traffic, uniform capacity– Optimal is direct routing between any two nodes (single-path)
General Traffic– Use Sprint’s 43-node fully-mesh telephone backbone network– Traffic for different time of the day– MPM∗: 19% on average– MPM∗: as high at 41%
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Fat-tree Data Center Networks
A special structure: k -pod architecture
NOT all pairs of nodes have traffic
Only Edge Switches have traffic intra-data center case
9 10 11 12 13 14 15 16
1 2 3 4 5 6 7 8
17 18 19 20
Edge
Aggregation
Core
k -pod fat-tree topologyN = 5
4 k2
D = k4
8 −k2
4
L = k3
2
L = O(k3),D = O(k4) MPM = L/D → 0
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Fat-Tree Topology: MPM and MPM∗ for LB
9 10 11 12 13 14 15 16
1 2 3 4 5 6 7 8
17 18 19 20
Edge
Aggregation
Core
k N D L MPM MPM∗ COH MPM∗ COH(U) (U) (LN) (LN)
4 20 28 32 100.00 35.71 14.29 15.00 0.536 45 153 108 69.93 39.87 5.88 10.85 0.368 80 496 256 51.41 28.02 3.23 9.47 0.31
U:=uniform traffic, LN:=lognormal traffic;COH:=Cost Overhead in (%)
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What happens when we start from a single demand pair andcontinue to add more demand pairs until D = N(N − 1)/2
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MPM* for grid, lognormal N = 16,25,36
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
X: 1Y: 3.664
Demand Type: Lognormal (µ =16.6, σ=1.04 )
No. of Demand Pairs / Total Pairs
MP
M*
(%)
Grid: 16 NodesGrid: 25 NodesGrid: 36 Nodes
[Warning: In some cases, single-path routing was not run longenough to reach optimality.]
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MPM* for grid, lognormal N = 49,64,100
0 0.1 0.2 0.3 0.4 0.50
10
20
30
40
50
60
X: 0.4354Y: 2.068
Demand Type: Lognormal (µ =16.6, σ=1.04 )
No. of Demand Pairs/Total Pairs
MP
M*
(%)
Grid: 49 NodesGrid: 64 NodesGrid: 100 Nodes
[Warning: In some cases, single-path routing was not run longenough to reach optimality.]
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MPM* and COH for N = 16,25,36 (Grid, lognormal)
#ofDemandPairsMPM* COH(%) #ofDemandPairsMPM* COH(%) #ofDemandPairsMPM* COH(%)1 100.00 100.00 1 80 69.79 1 100.00 113.892 70.00 117.86 2 60 43.04 2 60.00 100.004 65.00 52.83 4 50 42.82 4 35.00 22.898 37.50 30.34 8 35 35.06 8 27.50 55.1716 25.00 24.16 16 18.75 12.92 16 25.00 30.6332 18.75 15.94 32 16.25 20.66 32 13.13 16.0764 9.06 3.36 64 13.438 10.62 64 12.81 4.09
120 4.83 0.00 128 7.966 1.11 128 7.81 1.01256 4.532 0.00 256 5.08 0.00300 3.664 0.00 512 3.75 0.00
630 2.32 0.00
16Nodes 36Nodes25Nodes
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MPM* and COH for N = 49,64,100 (Grid, lognormal)
#ofDemandPairsMPM*(%) Overhead(%)#ofDemandPairsMPM*(%) Overhead(%)#ofDemandPairsMPM*(%) Overhead(%)1 60.00 23.33 1 40.00 14.29 1 20.00 6.672 50.00 46.51 2 20.00 24.24 2 10.00 21.744 35.00 57.89 4 35.00 40.00 4 30.00 4.178 20.00 20.35 8 15.00 15.79 8 12.50 0.0016 15.00 6.95 16 8.75 4.30 16 13.75 5.7132 11.25 9.27 32 13.75 13.27 32 14.37 8.0064 9.06 22.49 64 9.69 14.75 64 8.75 18.52
128 7.66 7.66 128 4.84 0.68 128 5.00 0.00256 5.31 0.10 256 3.59 0.16 256 5.31 1.40512 2.07 0.00 512 3.40 0.63 512 2.42 0.001024 1.78 0.00 1024 1.11 0.50 1024 1.62 2.151176 1.26 0.00 2016 0.92 0.00 2048 0.98 0.11
4096 0.40 0.004950 0.34 0.00
100Nodes49Nodes 64Nodes
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So, what went wrong in our thought process about thebenefit of multipath routing?
Our minds do play trick:)— We forget that others are using the network too— Remember the Manhattan Street Network
Often, smaller topologies were studied where multipath iscertainly beneficialFor large problems, heuristic algorithms were developed toshow the “benefit” of multipath routing– Problem is ....– Heuristic gives a false sense of benefit of multipathrouting since the solution is near optimal, but not anoptimal vertex solution!
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So, what went wrong in our thought process about thebenefit of multipath routing?
Our minds do play trick:)— We forget that others are using the network too— Remember the Manhattan Street NetworkOften, smaller topologies were studied where multipath iscertainly beneficial
For large problems, heuristic algorithms were developed toshow the “benefit” of multipath routing– Problem is ....– Heuristic gives a false sense of benefit of multipathrouting since the solution is near optimal, but not anoptimal vertex solution!
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So, what went wrong in our thought process about thebenefit of multipath routing?
Our minds do play trick:)— We forget that others are using the network too— Remember the Manhattan Street NetworkOften, smaller topologies were studied where multipath iscertainly beneficialFor large problems, heuristic algorithms were developed toshow the “benefit” of multipath routing
– Problem is ....– Heuristic gives a false sense of benefit of multipathrouting since the solution is near optimal, but not anoptimal vertex solution!
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So, what went wrong in our thought process about thebenefit of multipath routing?
Our minds do play trick:)— We forget that others are using the network too— Remember the Manhattan Street NetworkOften, smaller topologies were studied where multipath iscertainly beneficialFor large problems, heuristic algorithms were developed toshow the “benefit” of multipath routing– Problem is ....– Heuristic gives a false sense of benefit of multipathrouting since the solution is near optimal, but not anoptimal vertex solution!
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3-node Load Balancing example - revisit: optimal vs. near optimal solution
Minimize rsubject to
d12: x12 + x132 = 5d13: x13 + x123 = 10d23: x23 + x213 = 7c12: x12 + x123 + x213 <= 10c13: x132 + x13 + x213 <= 10c23: x132 + x123 + x23 <= 15End
1
3
2
Capaci
ty = 10
Capacity = 10
Capacity = 15
Traffic = 5
Traffic = 7Traffic =
10
Optimal r∗ = 0.75One demand is always single-path at optimality; In this case, pair 2:3 has single pathrouting, x23=7. Forcing this demand pair to take two paths will result in multipath forevery pair, but the solution is NOT optimal.Let’s say, x23 = 7− ε, and x213 = ε > 0. Then, the best r becomes 0.75 + ε, which isNOT optimal.
Objective
Feasible Region
Optimal Vertex
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Science and Engineering in Network Management
Are we always swayed by our drive to get a “better”approach?– Engineering
Are we forgetting to study a system as it is?– Science
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Science and Engineering in Network Management
Are we always swayed by our drive to get a “better”approach?– Engineering
Are we forgetting to study a system as it is?– Science
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Summary
Back to the original question: Is it a Panacea?It depends:)For networks N ≤ 25 with all pair traffic, it’s reasonablybeneficialThe benefit of multipath routing diminishes as N increasesand L = O(N) [realistic ISP topologies]If N ≈ 100, the benefit is quite minimal.MPM∗ observed is much lower than theoretical MPMThe objective function is not an impacting factorD + L result is traffic/capacity invariant.Science of a Network Management problem is important toinvestigate!
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Thank You!
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What about TCP Throughput?
TCP throughput problem is modeled as a utilitymaximization problem:
maxx≥0,X≥0
∑d∈D
wd log Xd
subject to∑
p∈Pd
xdp = Xd , d ∈ D∑d∈D
∑p∈Pd
δdp` xdp ≤ c`, ` ∈ L(7)
There are D + L constraintsWe can see that the objective is non-linear concave; wecan use the piece-linear approximation trick.That is, the D + L property holds
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X. Liu, S. Mohanraj, M. Pioro, and D. Medhi, “MultipathRouting from a Traffic Engineering Perspective: HowBeneficial is It?”, Proc. of 22nd IEEE InternationalConference on Network Protocols (ICNP), The ResearchTriangle, North Carolina, October 2014.http://sce2.umkc.edu/csee/dmedhi/papers/lmpm-icnp-2014.pdf
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